Well, there's absolutely nowhere that's safe outdoors during a thunderstorm. So that limits you a lot.
Normally I would suggest the Cabin Loop, but there may still be closures in effect from the low-intensity fire they had up there earlier this year, and the Mogollon Rim is usually a pretty active spot when it comes to rain and lightning during the monsoon.
The Huachucas are a great place to string together a couple of nights. Hamburg, Pat Scott, some Crest Trail over to Bathtub Spring with a side trip to Miller and Carr Peaks. But again, if you've got storms, being up in a sky island isn't ideal.
You might want to check out the Hualapai Mountains. It might be a bit far to drive from Tucson, but being farther out west might limit the storms. I think the full loop is only about 15 miles, so spreading it over three 5 mile days doesn't fill too much time. But it's nice up there and the views from the various peaks are pretty good.
Ja vi elsker dette landet